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Frequency Regulation with Storage: On Losses and Profits

Dirk Lauinger, François Vuille, Daniel Kuhn

Abstract

Low-carbon societies will need to store vast amounts of electricity to balance intermittent generation from wind and solar energy, for example, through frequency regulation. Here, we derive an analytical solution to the decision-making problem of storage operators who sell frequency regulation power to grid operators and trade electricity on day-ahead markets. Mathematically, we treat future frequency deviation trajectories as functional uncertainties in a receding horizon robust optimization problem. We constrain the expected terminal state-of-charge to be equal to some target to allow storage operators to make good decisions not only for the present but also the future. Thanks to this constraint, the amount of electricity traded on day-ahead markets is an implicit function of the regulation power sold to grid operators. The implicit function quantifies the amount of power that needs to be purchased to cover the expected energy loss that results from providing frequency regulation. We show how the marginal cost associated with the expected energy loss decreases with roundtrip efficiency and increases with frequency deviation dispersion. We find that the profits from frequency regulation over the lifetime of energy-constrained storage devices are roughly inversely proportional to the length of time for which regulation power must be committed.

Frequency Regulation with Storage: On Losses and Profits

Abstract

Low-carbon societies will need to store vast amounts of electricity to balance intermittent generation from wind and solar energy, for example, through frequency regulation. Here, we derive an analytical solution to the decision-making problem of storage operators who sell frequency regulation power to grid operators and trade electricity on day-ahead markets. Mathematically, we treat future frequency deviation trajectories as functional uncertainties in a receding horizon robust optimization problem. We constrain the expected terminal state-of-charge to be equal to some target to allow storage operators to make good decisions not only for the present but also the future. Thanks to this constraint, the amount of electricity traded on day-ahead markets is an implicit function of the regulation power sold to grid operators. The implicit function quantifies the amount of power that needs to be purchased to cover the expected energy loss that results from providing frequency regulation. We show how the marginal cost associated with the expected energy loss decreases with roundtrip efficiency and increases with frequency deviation dispersion. We find that the profits from frequency regulation over the lifetime of energy-constrained storage devices are roughly inversely proportional to the length of time for which regulation power must be committed.
Paper Structure (22 sections, 19 theorems, 64 equations, 11 figures, 2 tables)

This paper contains 22 sections, 19 theorems, 64 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Reduction to a Deterministic Optimization Problem
  4. Candidate Solutions
  5. Analytical Solution
  6. Case Studies
  7. Model Parametrization
  8. Marginal Cost and Maximum Regulation Bid
  9. Profits
  10. Conclusions
  11. Acknowledgements
  12. Proofs
  13. Supplementary Material
  14. Detailed Model Parameters
  15. Frequency Regulation
  16. ...and 7 more sections

Key Result

Proposition 1

All else being equal, the battery state-of-charge $y(x^b,x^r,\delta,y_0,t)$ is concave and strictly increasing in $x^b$, concave in $x^r$, concave nondecreasing in $\delta$, and affine nondecreasing in $y_0$.

Figures (11)

  • Figure 1: Distribution of the power flow $(\eta^+[\delta(t)]^+ - \frac{1}{\eta^-}[\delta(t)]^-)x^r$ entering the battery at any time $t$ for $x^b = 0$.
  • Figure 2: Feasible set and profit for $\eta^+ = \eta^- = 0.707$, $\Delta = 5 \cdot 0.0816$, $\gamma = 5$h, $T = 24$h, $\bar{y} = 100$kWh, $\bar{y}^+ = 6$kW, $\bar{y}^- = 6$kW, $c^r = 0.9$cts/kWh, $c^{b} = 4 c^r$.
  • Figure 3: Marginal increase in expected power loss for different roundtrip efficiencies.
  • Figure 4: Profit per unit of regulation power for wholesale and retail electricity prices.
  • Figure 5: The impact of charging and discharging losses on regulation power and profits.
  • ...and 6 more figures

Theorems & Definitions (46)

  • Remark 1
  • Proposition 1
  • Lemma 1
  • Proposition 2: Constraint reduction
  • Proposition 3: Properties of the expected terminal state-of-charge
  • Remark 2: Mean absolute deviation
  • Proposition 4: Implicit function
  • Lemma 2: Asymptotic slope
  • Lemma 3: Linearity
  • Remark 3: Computability
  • ...and 36 more